We aware of a rigorous proof [1]} that fourth order cumulants are subleading order terms in the expansion (REF ) under weak linearity for spatially homogeneous systems. We also appreciate that there are layers of subtractive cancellation in this problem. Our efforts suggest that what applies to a spatially homogeneous system of amplitude modulated interactions does not necessarily apply to a spatially inhomogeneous system of frequency modulated wavepackets energized by extreme scale separated interactions. This line of argument is further sustained by the finding in [2]} that the resonant bandwidth \(\Gamma \)  tends to the rms Doppler shift at finite amplitude. This result is aphysical and we have turned to path integral techniques to provide a resolution at the next order of time scales. It is at this order in the extreme scale separated limit that we envision promoting fourth order cumulants to leading order status.
